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January 27th, 2012, 03:28 PM   #1
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continuous

Suppose f:R->R and g:R->R are continuous functions, and define a new function h:R->R by h(x)=max{f(x),g(x)}. Is h continuous? Why?
I don't know how to begin; I think I should discuss different situation because f can be greater then g or smaller than g or they have intersection points.
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January 27th, 2012, 10:34 PM   #2
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Re: continuous

Well, consider the graphs of f and g on the same page. If one function is above the other, say f above g, then h = f and is continuous because f is continuous.

If for some x values g is above, for other x values f is above, they must intersect. The intersection will be the max and belong to h. The other parts of h will be from either f or g depending which is on top or other intersections.

Ether way h is continuous, though may be composed of different parts of f and g.

The point here is that if 2 continuous functions cross each other they must intersect in at least 1 point.



Another good question to ask about given the stated conditions...is h(x) differentiable everywhere? Why?
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January 31st, 2012, 02:19 AM   #3
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Re: continuous

Quote:
Originally Posted by frankpupu
Suppose f:R->R and g:R->R are continuous functions, and define a new function h:R->R by h(x)=max{f(x),g(x)}. Is h continuous? Why?
I don't know how to begin; I think I should discuss different situation because f can be greater then g or smaller than g or they have intersection points.

Take : f(x) = 0 ,which is a continuous function ,and g(x) = sinx , or g(x) = cos x,then the max{f(x),g(x)} are just point above the x- axis.

Can this function be continuous ??
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January 31st, 2012, 04:27 AM   #4
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Re: continuous

They are both continuous everywhere but not differentiable everywhere. You lose the derivative whenever they intersect.

if f(x)=0 then parts of the x axis belong to h(x), precisely those parts where g(x) <= 0

http://www.wolframalpha.com/input/?i=ma ... ual=Submit

http://www.wolframalpha.com/input/?i=ma ... ual=Submit
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January 31st, 2012, 05:17 AM   #5
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Re: continuous

Quote:
Originally Posted by outsos
Can this function be continuous ??
Yes, not all points are above x-axis.
Quote:
Originally Posted by outsos
Take f(x) = 0 ,which is a continuous function ,and g(x) = sinx
Gives h(x) = max{f(x),g(x)} = max{0, sin(x)}
Now, h(0) = max{0,sin(0)} is above the x-axis?
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